PROBABILITY 2026-05-10

Central Limit Theorem

Theorem

(Central Limit Theorem)

Let $X_1,X_2,\ldots$ be i.i.d. random variables with $\mathbb{E}[X_i]=\mu$ and $\operatorname{Var}(X_i)=\sigma^2<\infty$ . Then

\sqrt{n}\,\frac{\bar{X}_n-\mu}{\sigma}\xrightarrow{\mathcal{D}}\mathcal{N}(0,1)

as $n\to\infty$ , where $\bar{X}_n=\tfrac{1}{n}\sum_{i=1}^n X_i$ .

Remark

(Universality)

The CLT is remarkable in that the limiting distribution $\mathcal{N}(0,1)$ is independent of the original distribution of the $X_i$ — only the existence of finite variance is required.